Mathematics TASKS. GCSE to ASIA-level BRIDGING. tangent lines at a drffarent paint on the surface

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1 Mathematcs GCSE to ASA-eve BRDGNG TASKS tangent nes at a drffarent pant on the surface

2 Pease compete one ne from the task st eow. A students must compete the mdde task: worked examp :!car task *Topc st Factorsng Smutaneous Laws of ndces Quadratc Equatons Equatons Equaton of a ne Manpuatng Surds

3 The suject knowedge audt on the next two pages are a coecton of questons that w chaenge yo-ur mathematca sks. You w need to e confdent on these sks as they w e needed n your ASA-Leve course. There are a tota of 26 questons. You w need to compete a tota of 20 questons wth fu workng out shown. You have aso een provded wth one of the chapters you can expect to earn more aout n your course. Ths chapter s caed 'The noma Theorem'. Rea fe uses of the Bnoma Theorem e n engneerng, economcs, archtecture, physcs, weather forecast and many other feds. Your task s to use these notes to teach yoursef ths chapter. Use the exampes, questons and fnd other resources to hep you wth ths. You w e assessed on ths n Septemer.

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5 Fnd the vaue off:. A cynder has ase radus X cm and heght h cm. A cone has ase radus X cm and heght h cm. The voume of the cynder and the voume of the cone are equa. Fnd h n terms of X. Gve your answer n ts smpest form. _ On the grd, sketch the graph of y = cos X for 0' & X & 360' T~e dagram shows a sod wax Rearrange = - u v f to make u the suject of the formua. Gve your answer n ts smpest form. The cynder has ase radus hand heght 9x. The cynder s meted down and made nto a sphere of radus r. ABCD s a square. Fnd an expresson for r n terms of X P and D are ponts on the y-axs. A s a pont on the x-axs. PAB s a straght The equaton of the ne that passes through the ponts A and D s y = -2x + 5 Fnd the ength of PD A Q The dagram shows a sod cone and a sod hemsphere. The cone has a ase of radus X cm and a heght of h cm.. The hemsphere has a ase of radus X cm. The surface area of the cone s equa to the surface area of the hemsphere. A C Fnd an expresson for h n terms of X. -F_ cm -----t X 2 The dagram shows an equatera hange ABC wth sdes of ength 6 cm. P s the mdpont of AB Q s the mdpont O~AC. APQ s a sector of a crce, centre A. Cacuate the area of the shaded regon. Gve Your answer correct to 3 sgnfcant fgures. -.. E 0 22 Sove the equaton - - =! Here s a quadratc sequence. 2 x ~ The expresson for the nth term of ths sequence spd + qn Fnd the vaue ofp and the vaue ofq. AC=7cm. AB=3 cm.de= 9cm.. / A+3 cm -4 D. j 25. (-2, ), B(6,5) and C(4, k) are the vertces of a rght-anged trangeabc. Ange ABC s the rght ange. Fnd an equaton of the ne that passes through A and C. Gve your answer n.the form ay + x = c where 0, and c are ntegers Ange ABC = ange CBD = ange BDE = 900 AngeBDE = 48" Cacuate the ength of CD Gve your answer correct to 3 sgnfcant dam the co-o~c turnng pont ofthe graph ofy =x2 + OX

6 Dn.mnm;-r "nr thn kn--;- +hanrnm Thn h;nnrno thnnmm You can expand ( + X)" where n = 0,, 2,3,... EXPANSON COEFFCENTS ( + x)o = (+~)~~+ x (+~)~=+2x+x~ 2 (+~)~=+3x+3~~+~~ 3 3 (+~)~~+4~+6~~+4x~+x~ (+x)5=+5x+0x2+0x3+5x4+xs The coeffcents form a pattern known as Pasca's trange. Each coeffcent n the trange s the sum of the two coeffcents aove t., Pasca's Trange was pushed n 4 654, ut was known j ; to the Chnese and the Persans n the th century. -, Use Pasca's trange to wrte the expanson of ( + 2 ~ n ) ~ - ascendng powers of y. 5, - The coeffcents are, 6.5,20,5,6, (,! 8 E, ( + ( 2~))~ E + 6(2y) + 5(2~)~ + ZO(ZY)~ + 5(2~)~ + 6 (2~)~ + ( 2~)~ = +2y+60~+60~+2403p+92f'+64y6 Wrte down the 6th row of Pasca's trange. Use the expanson of ( + X)", susttutng 2y for X ; Repacng wth a and x wth gves the noma expanson (a + )" where n = 0,, 2,3,... As n ncreases you can see that agan the coeffcents form Pasca's trange. expresson has two terms. exampes (a + ) E of the ~non~~a expansan. (a (a + )3=

7 n each expanson, the power of a starts at n and decreases y each term, so the powers are n, n -, The power of starts at 0 and ncreases y each term, so the powers are 0,,2,..., n?he sum of the powers of any ndvdua term s aways n ' m Expand (2 + 3t)4 2 =24+4x23x(3t)+6x22~(3t)2+4x2x(3t)3+(3t)4 W =6+96t+26t2+26t3+8t a- -.-* Use Pasca's trange and the expanson of (a + )' susttutng 2 for a and 3tfor W p---!, t woud e mpractca to use Pasca's trange every tme you need to work out a coeffcent-'say, for exampe, you want to fnd the coeffcent of XG n (X + a)o There s a genera rue for fndng ths coeffcent wthout needng to wrte out Pasca's trange up to the tenth row. n: stands for the product of a ntegers from to n. ' You read t as n factora. : "C, s the choose functon and you read t as 'n choose L t gves! the numer of posse ways of choosng r eements from a set of n [ eements when the order of choosng does not matter. For exampe,! ' the numer of comnatons n whch you can choose 2 as from a ;- ag of 5 as s 5C2 C - YOU use the choose functon ecause there are severa ways of gettng certan powers from an expanson. For exampe, there are 3 ways of gettng a2 from the expanson of (a + 3: a from ether the frst, second or thrd racket and from the other two rackets n / each case. The term n a2 for the expanson of (a + ) s therefore j 3Ca2 = 3a2 J p----.aly-.-.-_ Note that the frst coeffcent n each row s the 0th coeffcent. GC- s sometmes wrtten as (:) or,,c. ' Look for the factora utton on your cacuator. t may e denoted X! ),

8 Povnomas and the noma theorem The noma theorem A term n the expanson of (y + s gven y ky3x6 Fnd the vaue of k Use your cacuator to 9C6xgx (2x)"=84x3x64x6 fnd OC, and work out 26 ~5376Yx6 Smpfy to fnd the k = 5376 vaue of k The formua for the noma expanson of (a + )"s sometmes caed the noma theorem. For the expanson of ( + X)" ths gves Wrte the term n 2 n the expresson (22 - )5. Smpfy your answer.! a Take a = 2zand =- so the second power! 5~~~(2~)5-(-) must e. Use the coeffcent Sc., Cacuate the vaues of a S! 7! c! 2 Cacuate the vaues of a 5C, 9C, c "C, d 3C8 3 Work out the vaues of a () (:D) C 5 Fnd the frst four terms of these noma expansons n ascendng powers of x a (+x)' (-3~)~ C (+2~)' d (2-3~) e (X-2)' f (2x- ) 6 Use Pasca's trange to expand each of (:3) (7) these expressons. a (2-4y) (3+5)' c (42-:) 4 Use Pasca's trange to fnd the expansons of each of these expressons. 7 Fnd the frst three terms of these noma expansons n descendng powers of x a ( + 3 ~ ) ~ (-;)' a (2+x)' (-2x)' c (-g C (3-4' d (~+4)~ e (2x+3) f (;+4J

9 8 Use the noma theorem to expand each of these expressons. 2 Expand and smphfy each of these expressons. a 3x(2~-5)' (2+~)~(+~) 3 Expand and smpfy each of these expressons. + (3 + 2x)' a (5-2 ~ ) ~ 9 Fnd the terms ndcated n each of these expansons and smpfy your answers. a (P + 55 term n p2 (4 +y9 term n $ C (3+q)2 term n q7 d (4-3m)5 term n m3 e (22 - )5 term n 9 g (3x + 4y)' term n y h (2a - 3) terms n a5 and 4 (4p+f) term n p2 j (4a-$) terms n a5 and 5 k (:-$) terms n a7 and 5 0 Use the noma theorem to expand each of these expressons. 4 Expand and fuy smpfy each of these expressons. Show your workng. a [2+~)~+(-&Y [- - (2& Wrte down the frst four terms of the expanson of each of these n ascendng powers of X a (+2x)" (-3x)" 6 a Expand (+ 4 ~ n ascendng ) ~ powers of x up to and ncudng the term n 2 Use your answer to part a to estmate the vaue of (.04)~ 7 a Expand ( -2~)~ n ascendng powers of X up to and ncudng the term n x3 Use your answer to part a to estmate the vaue of (0.99)~ 8 Use the noma expanson to smpfy each of these expressons. Gve your fna soutons n the form a + f a (+&) (-&) Use the noma theorem to expand each, of these rackets. 9 Use the noma expanson to fuy smpfy each of these expressons. Gve your fna answers n surd form.

10 -P Pnvnnmnc nnr tp hnnma thenrpm T ~ hnnrnn P Create an expresson n the form ( + X)" or (a + Use Pasca's trange or the noma theorem to fnd the requred terms of the noma Use your expanson to answer the queston n context. K; n CL ' E {! A foota squad conssts of 3 payers. Use the formua "C, = to show that there are (n-r)!r! 78 posse comnatons of choosng a team of payers from ths squad. m' j! an" 3 = (3- )!! 3x2~ xox... x2x -! 2!x X 0x... x 2x 3x2-2! Cance the common factor! ( a Usng the frst three terms of the noma expanson, estmate the vaue of.003' - a, By cacuatng the'fourth term n the expanson show that the estmate from part a s D. accurate to 3 decma paces. E a.0038 = ( )' Rewrte n the form ( +A) Frst 3 terms f! "("-x2. = +nx+- Use the frst 3 terms of 2! genera expanson. = = (=.024to 35f) Susttute vaues and "("-)(n-2) X 3 = 56(0.003)~ 3! smpfy. - Addng ths term w not affect the frst three decma paces.

11 How many posse ways are there to pck a 7's rugy team from a squad of 0 payers? 4 n the expanson of +- a postve P 2 How many posse ways are there to choose haf of the peope n a group of 20? - 3 A cue has sde ength (2s- 3w). Use the noma expanson to fnd ts voume. Use Pascas trang fnd the vaue of p a.05= correct to sx decma paces.96' correct to four decma paces. 5 Use the noma theorem to work out the vaue of a.055 correct to 4 decma paces, (g)'' correct to L sgnfcant fgmes. 6 Use the noma theorem to work out the (3 vaue of - correct to fve decma paces. ; 7 Work out the exact vaue of the mdde term ' n the expanson of (&+&)0 8 a Fnd the coeffcent ofx4 n the expanson of ( + X ) (2x - 3)5 Fnd the coeffcent of x3 n the expanson of (X - 2) (3x + 5)4 9 Fnd, n the expanson of coeffcent of 0 Fnd, n the expanson of coeffcent of f me frst three termsn the expanson of (+ax)" are +35x+490x2. Gven that n s a, postve nteger fnd the vaue of <. a n a 2 Gventhat(+xr =-24x+252xZ+ for a 7 postve nteger n fnd the vaue of ' 8 '3 n the expanson of (+2x): n a postve 6. nteger, the coeffcent of x2 s eght tmes 6. the coeffcent of X. Fnd the vaue of n A' ( 3 nteger, the coeffcents of x4 and x5 are equa. Cacuate the vaue of n 5 Fnd an expresson for Wrte your answers as poynomas n n wth smpfed coeffcents. 6 Fuy smpfy these expressons. n! (n + 3)! a - - (n + )! n(n+)! 7 Fnd the constant term n the expanson of (2 + 3x)3 ($ - 4r 8 Fnd the coeffcent of y3 n the expanson of (Y + 53 (2 - Y 5 Chaenge 9 A test nvoves 6 questons. For each queston there s a 25% chance that a,student w answer t correcty. a How many ways are there ' of gettng exacty two of the questons correct? c d What s the proaty of gettng the frst two questons correct then the next four questons ncorrect? What s the proaty of gettng exacty two questons correct? What s the proaty of gettng exacty haf of the questons correct?

12 You can expand ( + X)" where n = 0,,2,3,... ( +x)o= ( +X)'= + x EXPANSON (+~)~=+2x+ x2 (+~)~=+3~+3x~+x~ COEFFCENTS The coeffcents form a pattern known as Pasca's trange. Each coeffcent n the trange s the sum of the two coeffcents aove t. was pushed n 654, ut was known to the Chnese and the Persans n the Use Pasca's trange to wrte the expanson of ( + 2 ~ n ) ~ ascendng powers of y. The coeffcents are, 6,5,20,5,6, ' ( + (2~))" E + 6(2y) + 5(2~/)~ + ZO(ZY)~ + 5(2~)~, +2y+60f+60~+2403p+92y5+64~ + 6(2y5 + ( 2 ~ ) ~ Wrte down the 6th row of Pasca's trange. Use the expanson of ( + X)", susttutng 2yfor X Repacng wth a and xwth gves the noma expanson (a + )" where n = 0,,2,3,... As n ncreases you can see that agan the coeffcents form Pasca's trange. expresson has two terms. m (a+)oc For ore exampes (a + ) E o te nona a+ expanson. (a + ' Z ' a2 + 2a + 2 (a + 3= a3 + 3a2 + 3a2 + 3

13 .--U. n each expanson, the power of a starts at n and decreases y each term, so the powers are n, n -, n-2,..., 0 The power of starts at 0 and ncreases y each term, so the powers are 0,,2,..., n The sum of the powers of any ndvdua term s aways n Expand (2 + 3t)" Use Pasca's trange and the expanson of (a + )4 :: susttutng 2 for a and B =24+4~23x(3t)+6~22~(3t)2+4x2x(3t)3+(3t)4 3tfor ~6+96t+26t2+26t3+8t4 t woud e mpractca to use Pasca's trange every tme you need to work out a coeffcent-say, for exampe, you want to fnd the coeffcent of 9 n (X + a) frst coeffcent n each row s the 0th There s a genera rue for fndng ths coeffcent wthout needng to coeffcent. out Pasca's trange up to the tenth row S f "Cr s sometmes n! 3 rth coeffcent n the nth row s "Cr = (n-r)!r! j wrtten.s (") R k or "C r A--. ~, tands for the product of a ntegers from to n. - ~ookfor the read t as n factora. : factora utton on For exampe,6!=6x5x4x3x2x=720 h your cacuator. t r :,. may e denoted X! L "C, s ; the choose functon and you read t as 'n choose r: t gves the r umer of posse ways of choosng r eements from a set of n eements when the order of choosng does not matter. For exampe, the numer of comnatons n whch you can choose 2 as from a ag of 5 as s 5C, You use the choose functon ecause there are severa ways of gettng certan powers from an expanson. For exampe, there are 3 ways of gettng a2 from the expanson of (a + )3: a from ether the frst, second or thrd racket and from the other two rackets n each case. The term n a2 for the expanson of (a + )3 s therefore 3C, a2 = 3a2 ' ' z.-7

14 Pnvnnmas and the hnnma theorem The hnom theorem A term n the expanson of (y + 2 ~ s ) gven ~ y ky3x6 Fnd the vaue of k! Use your cacuator to / 9C6xy'x ( 2 ~ ) ~ = 0 4 ~ ~ ~ 6 4 x ~ fnd 9C, and work out 26 g L! k = 5376 vaue of X ( The formua for the noma expanson of (a + )"s sometmes caed the noma theorem. (a + )" = an + "Can- + "C,U"-~~~ "Cpn-'Lf " For the expanson of ( +X)" ths gves Wrte the term n 2 n the expresson (225 - )5. Smpfy your answer. Take a = 2zand =- 5C,(2z)5-'(-) * 365~ 69~(-) The powers add to 5 so the second power must e. Use the coeffcent 5C E (2~)~ E 24~4 Cacuate the vaues of 5 Fnd the frst four terms of these noma expansons n ascendng powers of x a S! 7! c! 2 Cacuate the vaues of a (+x)' (-3x)' C (+2~)' d (2-3~) a 5C, 9C, c "C, d "C, 3 e (X-2)'' f (2x- ) Work out the vaues of 6 Use Pasca's trange to expand each of a ( ) ( ) d ( ) theseexpressons. / 3, \5 a (2-4y) (3 + 5)' 4 Use Pasca's trange - to fnd the expansons of each of these expressons. 7 Fnd the frst three terms of these noma expansons n descendng powers of X a ( + 3 ~ ) ~ [-;)5 a (2+~)~ (-2x)' C (-:)( [+$)5 c (3-x~ d (x+~)~ e (2x+3y0 f (:+4T C (42 - $)

15 ' Use the noma theorem to expand each of these expressons. a (2+3t4 (3-2p4 2 Expand and smpfy each of these expressons. a 3x(2~-5)~ (2+~)~(+~) 3 Expand and smpfy each of these expressons. a (5-2 ~ + (3 ) + ~ 2x) Fnd the terms ndcated n each of these expansons and smpfy your answers. a (P + 55 term n (4 + Y ) ~ term n f C (3+q)2 term n q7 d (4-3m)5 term n m3 e (22 - )5 term n 2 g (3x + 4 ~ ) term ~ n y h (2a- 3) terms n a5 and 4 3 (.p+$) term n p2 j (4a-$) terms n a5 and V ; 0 Use the noma theorem to expand each of these expressons. a (L~+cP)~ (3 - w')~ C (2s2 + 5P)3 d (2s'- 4 Expand and fuy smpfy each of these expressons. Show your workng. a (2+,6)4+(-&)4 (- &) - (2& + 3 r 5 Wrte down the frst four terms of the expanson of each of these n ascendng powers of x a (+2x)" (-3x)" 6 a ~xpand ( + 4 ~ n ascendng ) ~ powers of x up to and ncudng the term n 2 Use your answer to part a to estmate the vaue of (.04)~ 7 a Expand (-24' n ascendng powers of xup to and ncudng the term n x3 Use your answer to part a to estmate the vaue of (0.99)~ 8 Use the noma expanson to smpfy each of these expressons. Gve your fna soutons n the form a + & a (+&)' (-6) Use the noma theorem to expand each, of these rackets. 9 Use the noma expanson to fuy smpfy each of these expressons. Gve your fna answers n surd form. a (+&r (-&) C (5-f) d (2&+5) e (&+C)' ' f (6-6)"

16 .?.L Poynomas and the noma theorem The noma Create an expresson n the form ( + x ) or ~ (a + Pasca's trange or the noma theorem to fnd the requred terms of the noma Use your expanson to answer the queston n context. n!, A foota squad conssts of 3 payers. Use the formua "C, = to show that there are -- (n-r)!r! - 78 posse comnatons of choosng a team of payers from ths squad. c- L >- - 3x2~ x0x... x2x 2!x X O X... x2x. 3x2 Cance the common = ! factor! , 2 B 'CO. a Usng the frst three terms of the noma expanson, estmate the vaue of.003a!?! By cacuatng the fourth term n the expanson show that the estmate from part a s (?- 3 accurate to 3 decma paces. -, :. 5!?< " a a.0038 = ( )~ L.- Frst 3 terms n(n - ) = +nx+- x2 2! L Rewrte n the form ( + x) Use the frst 3 terms of the genera expanson. r = + a(0.003) + 20(0.003)' = = (=.024to 3 sf) n(n-)(n-2) X 3 = 56(0.003)~ = Addng ths term w not affect the frst three decma paces. Susttute vaues and smpfy. 3!

17 S How many posse ways are there to pck a 7's rugy team from a squad of 0 payers? ( 3 2 How many posse ways are there to choose haf of the peope n a group of 20? ". 3 A cue has sde end (2s- 3w. Use the noma expanson to fnd ts voume. 4 Use Pasca's trange to fnd the vaue of a.056 correct to sx decma paces.963 correct to four decma paces. 5 Use the noma theorem to work out the vaue of a (E)".055 correct to 4 decma ~aces. correct to he sgnfcant fgures. (3. 6 Use the noma theorem to work out the vaue of - correct to fve decma paces. 7 Work out the exact vaue of the mdde term. n the expanson of (&+&)0 8 a Fnd the coeffcent ofx4 n the expanson of ( + X) (2x - 3)5 Fnd the coeffcent of x3 n the expanson of (X - 2) (3x + 5)' 9 Fnd, n the expanson of coeffcent of 0 Fnd, n the expanson of ($ + t3 r coeffcent of the The frst three terms n the expanson of (+ax)" are +35x+490x2. Gven that n s a postve nteger fnd the vaue of 2 Gventhat(+x)"-24x+252x2+... fora postve nteger n fnd the vaue of.3 n the expanson of (+2x)", n a postve nteger, the coeffcent of x2 s eght tmes the coeffcent of X. Fnd the vaue of n 4 n the expanson of + - n a postve nteger, the coeffcents of x4 and xs are equa. Cacuate the vaue of n 5 Fnd an expresson for Wrte your answers as poynomas n n wth smpfed coeffcents. 6 Fuy smpfy these expressons. n! (n + 3)! a (n + )! n(n+)! 7 Fnd the constant term n the expanson of 8 Fnd the coeffcent of y3 n the expanson of (U + 53 (2 - Y 5 Chaenge 9 A test nvoves 6 questons.,. For each queston there s a 25% chance that a,student w answer t correcty. a HO& many ways are there of gettng exacty two of the questons correct? What s the proaty of gettng the frst two questons correct then the next four questons ncorrect? c d What s the proaty of gettng exacty two questons correct? What s the proaty of gettng exacty haf of the questons correct?